Bounded set (topological vector space)


In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set.
A set that is not bounded is called unbounded.
Bounded sets are a natural way to define locally convex polar topologies on the vector spaces in a dual pair, as the polar set of a bounded set is an absolutely convex and absorbing set.
The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935.

Definition

Suppose is a topological vector space over a topological field
A subset of is called ' or just ' in if any of the following equivalent conditions are satisfied:

  1. : For every neighborhood of the origin there exists a real such that for all scalars satisfying
  2. is absorbed by every neighborhood of the origin.
  3. For every neighborhood of the origin there exists a scalar such that
  4. For every neighborhood of the origin there exists a real such that for all scalars satisfying
  5. For every neighborhood of the origin there exists a real such that for all real
  6. Any one of statements through above but with the word "neighborhood" replaced by any of the following: "balanced neighborhood," "open balanced neighborhood," "closed balanced neighborhood," "open neighborhood," "closed neighborhood".
  7. For every sequence of scalars that converges to and every sequence in the sequence converges to in
    • This was the definition of "bounded" that Andrey Kolmogorov used in 1934, which is the same as the definition introduced by Stanisław Mazur and Władysław Orlicz in 1933 for metrizable TVS. Kolmogorov used this definition to prove that a TVS is seminormable if and only if it has a bounded convex neighborhood of the origin.
  8. For every sequence in the sequence converges to in
  9. Every countable subset of is bounded.
If is a neighborhood basis for at the origin then this list may be extended to include:

  1. Any one of statements through above but with the neighborhoods limited to those belonging to
    • e.g. Statement may become: For every there exists a scalar such that
If is a locally convex space whose topology is defined by a family of continuous seminorms, then this list may be extended to include:

  1. is bounded for all
  2. There exists a sequence of non-zero scalars such that for every sequence in the sequence is bounded in .
  3. For all is bounded in the normed space">Normed space">normed space
  4. B is weakly bounded, i.e. every continuous linear functional is bounded on B
If is a normed space with norm, then this list may be extended to include:

  1. is a norm bounded subset of By definition, this means that there exists a real number such that for all

  3. is a subset of some ball.
    • This ball need not be centered at the origin, but its radius must be positive and finite.
If is a vector subspace of the TVS then this list may be extended to include:

  1. is contained in the closure of
    • In other words, a vector subspace of is bounded if and only if it is a subset of
    • Recall that is a Hausdorff space if and only if is closed in So the only bounded vector subspace of a Hausdorff TVS is
A subset that is not bounded is called.

Bornology and fundamental systems of bounded sets

The collection of all bounded sets on a topological vector space is called the or the
A or of is a set of bounded subsets of such that every bounded subset of is a subset of some
The set of all bounded subsets of trivially forms a fundamental system of bounded sets of

Examples

In any locally convex TVS, the set of closed and bounded disks are a base of bounded set.

Examples and sufficient conditions

Unless indicated otherwise, a topological vector space need not be Hausdorff nor locally convex.
Unbounded sets
A set that is not bounded is said to be unbounded.
Any vector subspace of a TVS that is not a contained in the closure of is unbounded
There exists a Fréchet space having a bounded subset and also a dense vector subspace such that is contained in the closure of any bounded subset of

Stability properties


  • In any TVS, finite unions, finite Minkowski sums, scalar multiples, translations, subsets, closures, interiors, and balanced hulls of bounded sets are again bounded.
  • In any locally convex TVS, the convex hull of a bounded set is again bounded. However, this may be false if the space is not locally convex, as the Lp space spaces for have no nontrivial open convex subsets.
  • The image of a bounded set under a continuous linear map is a bounded subset of the codomain.
  • A subset of an arbitrary (Cartesian) product of TVSs is bounded if and only if its image under every coordinate projections is bounded.
  • If and is a topological vector subspace of then is bounded in if and only if is bounded in
    • In other words, a subset is bounded in if and only if it is bounded in every topological vector superspace of

Properties

A locally convex topological vector space has a bounded neighborhood of zero if and only if its topology can be defined by a seminorm.
The polar of a bounded set is an absolutely convex and absorbing set.
Using the definition of uniformly bounded sets given below, Mackey's countability condition can be restated as: If are bounded subsets of a metrizable locally convex space then there exists a sequence of positive real numbers such that are uniformly bounded.
In words, given any countable family of bounded sets in a metrizable locally convex space, it is possible to scale each set by its own positive real so that they become [|uniformly bounded].

Generalizations

Uniformly bounded sets

A family of sets of subsets of a topological vector space is said to be in if there exists some bounded subset of such that
which happens if and only if its union
is a bounded subset of
In the case of a normed space, a family is uniformly bounded if and only if its union is norm bounded, meaning that there exists some real such that for every or equivalently, if and only if
A set of maps from to is said to be if the family is uniformly bounded in which by definition means that there exists some bounded subset of such that or equivalently, if and only if is a bounded subset of
A set of linear maps between two normed spaces and is uniformly bounded on some open ball in if and only if their operator norms are uniformly bounded; that is, if and only if
Assume is equicontinuous and let be a neighborhood of the origin in
Since is equicontinuous, there exists a neighborhood of the origin in such that for every
Because is bounded in there exists some real such that if then
So for every and every which implies that Thus is bounded in Q.E.D.
Let be a balanced neighborhood of the origin in and let be a closed balanced neighborhood of the origin in such that
Define
which is a closed subset of that satisfies for every
Note that for every non-zero scalar the set is closed in and so every is closed in
It will now be shown that from which follows.
If then being bounded guarantees the existence of some positive integer such that where the linearity of every now implies thus and hence as desired.
Thus
expresses as a countable union of closed sets.
Since is a nonmeager subset of itself, this is only possible if there is some integer such that has non-empty interior in
Let be any point belonging to this open subset of
Let be any balanced open neighborhood of the origin in such that
The sets form an increasing cover of the compact space so there exists some such that .
It will be shown that for every thus demonstrating that is uniformly bounded in and completing the proof.
So fix and
Let
The convexity of guarantees and moreover, since
Thus which is a subset of
Since is balanced and we have which combined with gives
Finally, and imply
as desired. Q.E.D.
Since every singleton subset of is also a bounded subset, it follows that if is an equicontinuous set of continuous linear operators between two topological vector spaces and, then the orbit of every is a bounded subset of

Bounded subsets of topological modules

The definition of bounded sets can be generalized to topological modules.
A subset of a topological module over a topological ring is bounded if for any neighborhood of there exists a neighborhood of such that